In the new apartment, the cat's litter box is in the bathroom. Her new hobby is to weird me the heck out by following me in and watching me go to the bathroom WHILE SHE GOES TO THE BATHROOM TOO. I'll give her bits of my morning cereal, but this is just too much sharing.

So all of our stuff is in the new place, which is FABULOUS, and save one mishap involving our couch being smaller than the front door, all went well. Right now there's a lot of furniture adjusting and what-should-we-do-with-this-odd-corner and making lists to bring to the nearest hardware-containing box store. And at the same time, we're having tons of visitors this week, which is lovely, as they're not the type to get judgemental about the empty boxes lining the hallway or the fact that we have nowhere to put towels yet.

The cat is adjusting well. She was way more freaked out, and for longer, than she was when we first brought her home from the SPCA. I think at this point she has a cushy life and a routine, so being removed from that really threw her for a loop. But now she's figured out where the good sunbeams and squirrel-watching spots are. The real turning point for her seemed to be when she figured out something which we never would have: that the previous tenants cared for a cat who liked to knock toys under the stove, but didn't really care to retrieve them. Punky, meanwhile, loves sticking her paw under things looking for fun. Four jingly mice, a catnip-stuffed hot dog, some twist ties, and many pipe-cleaner-springs later, she lay spent and content on the kitchen floor, surrounded by incredibly dusty treasures.

Clearly there's something in this apartment for every resident.

So we've decided to move into a new apartment in the most last-minute manner possible. We decided to see if we could move last Tuesday, we signed a lease this Monday, we're moving our stuff on Saturday.

This is awesome, I love the new apartment, but now I dream of boxes, and of Tetrising all my stuff into them.

Our particular version of soppy romance:

I bring J's wedding ring to work with me so that I can electrochemically clean it.(They're silver, they tarnish over time.)

J sees a 1923 copy of Einstein's 1921 Princeton lectures in a pile of books that the library is discarding, and brings it home to me.

We are each filled with warm tender feelings as a result. I might have actually sad "aw" when I read the binding.

So before I started writing the blog posts about how to get equal stripes in a triangular shawl, I started spinning the yarn I wanted to use in a triangular shawl. And finished it.



This is most of it--an alpaca-merino-silk blend from Susan's Spinning Bunny, in the "blueberry patch" colorway, laceweight singles, about 25wpi. I don't normally name yarns but the colors originally attracted me because the ball of fiber looked like some kind of funky globe, so I'm calling it Geography Bee (close runner up: Class M, as the globe was clearly not Earth. But even I am not enough of a geek to pull that naming scheme off). There's less than 1/2 ounce left to spin. It could have easily fit into one skein, but the Joy just isn't up to the task of spinning this at the speed and intake rate I wanted with a mostly-full bobbin. That just means some careful labeling so I know which end of which skein to start with.

I realized after the end of the first blog post that I'd made a terrible mathematical mistake when I spun the fiber up. Just so dumb. And since then, I've both been A: embarrassed that I didn't realize the problem, and B: having a difficult time of figuring out how to explain it.

I think I'm there. Bear with me.

So when I first calculated out these relative lengths, all I was thinking about was if things increased in a linear way, or with x squared. Either made sense before I sat down with some paper. I got far enough to figure out that it was linear (so not quite even to the end of the second blog post) and left it there. This would be easy! I would just break my fiber into thinner strips as I went, in a linear way, making each length of color shorter.

I broke the fiber widthwise into roughly 1/3 ounce chunks--12 pieces all together. There were 3-5 handdyed colors in each chunk. This would mean that there was some variability in stripe width, but I wasn't looking for perfection, just stripes that didn't start off as one huge wedge and end as skinny little things.

"Hmm," I thought to myself, "the increases in length happen in a linear way. Spinning the first chunk without doing any splitting will result in the widest possible stripes. So if I start with one whole piece, then split the fiber widthwise into two pieces, then three, four, etc, it will work out, more or less."

Two problems. The more minor one is that, as we figured out in the last entry, each single-color length needs to increase as *2x*, not x. I was thinking 1,2,3,4, not 1,3,5,7. I had actually increased at that rate, I would have ended up with a similar effect to what I was trying to avoid, just slower. Not ideal.

However, the larger problem is that splitting things up into 2,3,4,etc *pieces* is not even linear to begin with. It's 1/x, not -x. If I were to measure the average length of a single color over several of these splits and graph it, what I'd end up with would look like this:

(Note to graphing-program geeks: Yes, Excel. I use it as my sandbox. Calm down.)

The blue line would show a colorway where the lengths actually decreased linearly, the pink line shows what I was doing.

Not. Even. Close. Now, I know this stuff and am usually pretty comfortable with my ability at this kind of thing. Ratios, relative amounts of things when switching units around, these are things that you need to do every day if you're doing experimental science. This was just a seriously dumb mistake.

Oddly, due to some variation in technique and also because there was something niggling at my brain that I was doing this WRONG (which caused this series of posts, which caused me to figure it out and slap myself in the forehead), I made some last minute modifications which slightly fixed the damage. But only slightly. Here's an estimate of what the lengths of individual colors will be along the length of the entire yarn, along with what it should have been:

Oddly, the starting and ending points are pretty darn close, certainly within the variability given by the fiber itself. It's that middle part that's all fakakte.

So what's it going to look like in the end? Still better than what it would be if I didn't vary the width of the fiber at all (that would be a flat line across the bottom, every section of color the same length.) The stripes ought to start at a medium width, narrow down a bit but not as much as they normally would, and then widen up again towards the bottom edge of the shawl. The last few stripes ought to be about the same width as the first few. I could calculate this and draw up an approximate picture, but at this point I am more excited about knitting the project than I am about continuing this exercise. Wouldn't you be? Just look at that picture up top. Gorgeous colors.

It will be interesting to take a picture of the final project and compare it to pictures of shawls that have been knit in these linear long-color-progression yarns. I suspect that it will be clear that something different is happening even to the casual observer, though exactly *what* is happening will probably be clear only to me and whatever other nut decides to do the math.

A tiny voice inside my head pipes up and asks me what I'll do next time, now that I have this knowledge (stupid voice, wreck my fun). The tricky part is that it's simply not *possible* to split a single chunk of fiber such that the weight increases linearly. Not in a way that's convenient for the crafter, at least. I have some ideas, but I'll let this particular loose end dangle for anyone else who would like to continue this process. Go to town, and let me know what you come up with.

Pointer to some amazingly beautiful art yarn: here. That gets all kinds of pistons firing. So lovely and also technically impressive.

So, as I was saying. Most triangular shawls are knit from the center back outwards, so each row is longer than the one before. This means that one of these shawls, when knit with stripes of equal yardage, end up with a wide triangle in the center back and stripes of ever decreasing thickness as you move towards the outer edge, like so:

This look just doesn't do it for me. If you're working with a commercial yarn like Noro or Kauni, then you have to deal with what you're given and I get that. But the whole point of spinning my own yarn is that I'm not limited by the choices that a far-off company has made. I can knit with exactly the yarn I want to knit, and for me, that involves color progression as surely as it does thickness, plies, fiber quality, etc, etc. (See also: Huntington Castle Pullover.)

If my final goal is a shawl with more-or-less equal stripes, like this:
How do I spin a yarn that does that?

Before the math, I need to make some guesstimates to make the math easier. I think that these assumptions are reasonable--hopefully no spherical cows. And anyway, I'm not nearly as concerned about absolute perfection as the algebra and geometry I'm about to get into would indicate. I don't mind some variability, I just don't want a huge honking triangle at the beginning and weeny little stripes at the end. Perfection is left as an exercise to the reader. :)

First, what I'm really interested in finding out is the length of yarn I'd need to make stripes of equal width, but I haven't swatched (I don't even have the YARN yet, just a ball of fiber), so I don't know exactly how much yarn covers how much area. So I'm going to assume that the same length of yarn will always cover the same area. This works as long as I increase evenly (so no part of the knitting gets distorted), and as long as I spin and knit evenly, without too many tight or loose spots. So instead of trying to convert things to yardage using any more formulas, I can just look at how the area of each stripe increases relative to the previous stripes. That's high school geometry and maybe a bit of algebra, we can handle that.

I'm also going to assume that the bottom point forms a perfect right triangle--in other words, that this shawl is exactly the same dimensions as a square chopped along the diagonal. This *should* be reasonably close based on what of these shawls I've seen before. This keeps us to only dealing with first semester high school geometry and algebra. Phew.

Now, those stripes. I'm going to call the width of each stripe along the top x, and I'm going to say that I have n number of them. (n=5 in the picture above, but I'm not knitting a shawl with 5 stripes, and I want this to be generalizable for any striped triangle pattern.)

The innermost blue triangle conveniently covers the nice round area of x². (Math review: Area=1/2 (base * height). The base of the triangle is 2x, because we're counting the area of the whole blue thing. The height is x, because we're assuming this is just a square hacked in two. 1/2 (2x * x)=x².)

The green stripe, it kind of looks like a triangle with a triangular notch cut out of it, right? I'm going to calculate the area of that stripe by finding the area of the triangle made by the blue and green together, then subtracting out the blue stripe (which I already know the area of.)

Area _blue+green=4x². (4x*2x, divided by 2.)

Area _green=4x²-x²=3x²

And I can keep going for the third, fourth, fifth stripes:

Area ₃= Area ₁₊₂₊₃ - Area ₁₊₂=9x²-4x²=5x²

Area ₄=16x²-9x²=7x²

Area ₅=25x²-16x²=9x²

Okay, the pattern is showing up. x², 3x², 5x², 7x², 9x². Each stripe needs two units more area than the one preceeding. We're assuming the same length of yarn covers the same length of area, so the increase in length for each stripe is linear. If the first bit of yarn was 1 inch long, the second bit would be 3 inches, then 5,7,9,11, and on and on like that. I just need to pick a unit, now, and increase more or less smoothly from there.

Now, I just have to figure out how to fit that math into the very real and lovely piece of hand-dyed fiber that I have. Tomorrow.

As anyone who is paying attention has noticed, I tend to think about craft projects with attention to ridiculous levels of mathematical detail. A sloppy seam doesn't bother me, but a pattern repeat that doesn't quite fit? Watch out!

So here's what I've been focusing on recently: striping in triangular shawls.

Let's say you want to knit a triangle shape to put over your shoulders. Any isosceles will do but I'm going to assume a right triangle for ease of computation later:



You can actually knit this in several ways: start from the bottom point and increase, or knit the thing from side-to-side by starting from one of the other corners. You could even cast on half of the top width, do some clever things with short rows, and end up casting off the other half of the top at the end. But the vast majority of patterns of this type have you start or end along the entire bottom edge, and then, end or start with just a few stitches at the back of the neck. Here are a few examples of this type of construction (the first is a long-edge-up, the other three are a center-neck-down). If you're working from the center neck down, there are increases along the center and on each side, and when you add in some patterning, you end up with a neat symmetry along the spine of the shawl, like so:

Except, you know. Looking nice, and not like I spent less than two minutes with MS Paint.

Now, let's say that you have several balls of yarn--all the same size but in different colors. Or, let's say you have one of those fancy yarns that changes color every X number of yards. The rows at the outer edge of the shawl are going to be much longer than the rows at the center top of the shawl. So if you use up each ball of yarn before going on to the next, or if your yarn changes color every 10 yards and you have no control over it, you'll end up with something that looks like this:



A big chunky not-even-stripelike-thing in the center, and ever-thinnening stripes down towards the edges.

I HATE this look. Sometimes it doesn't bother me so much, like when the project is just a cozy stashbuster, but for something that is supposed to be a source of pride, no. Just no. Especially when I'm spinning for a project and thus have complete control over the final yarn, it just seems terrifically lazy.

I've got some very pretty hand-dyed fiber that I'd like to make into a triangular shawl. Can I split it up in a way that eliminates, or at least minimizes, this stripey effect?

You bet I can. Next time: some actual math.